A. C. I>IXON ON S-ITKM UOI'VII.I.K ll.\i;M"\l< I.Xl'ANsIuNS 



431 



constant \alue s<> long as those intervals do not change, and the corresponding part 

 of tin- double integral is a constant multiple of 



In tliis expression take tin- Irniis of',,.(/,.r) separately. Taking tli' coefficient of K 



we lia\f 



, 0)- \ * (* 0)+ 1 



A A.r,, 



CC; -*( ap .)*('> i )<fc<fe 



,o) 



i f ' 



A.'o 



wliich is at most of the order of RT 32 ^ (1,0) so long as JT,, < Xi : t a < ^i. In this way 

 it appecars that the triple integral is at most of the order of R" 1 *, and tin- same result 

 can be deduced when t H < , S x<t < Xi by putting w (r, t) in the place of -.< (/, a-). Since 

 the number of intervals is finite the whole expression (21) tends to zero when 11 is 

 increased indefinitely, and the integral (20) can be made as small as we please when 

 f(x) is a function of the special type, constant in each of a system of sub-intervals. 



Now, letyX*) " )e unrestricted but real, 0(x) a real function of the special type, and 

 X (x) such an expression as 



"i (r) + a^t (j-) +...+ <7 w f w (.r). 

 We have 



f - \fa- x x\ a dx < 2 f - (fx-jxYdx+2 f \tx- x x\'dx, 



Jo p Jo p Jo p 



and it follows from 5, 6 that the first of these terms can be made arbitrarily small, 



if the integral of - (fa)* exists, by proper choice of the constant values of tf> and of 



P 

 the different sub-intervals. It has now also l)een proved that the second term can IM- 



diminished indefinitely by a proper choice of a t , fij, ... and by taking in great enough. 



f 1 1 



Hence \Jx-yX\'dx t or (20), is also arbitrarily small. Now (20) has its minimum 

 Jo p 



value for any given value of m when i, ..., have the values assigned in 2:}, and 

 this minimum value must tend to zero when m is. increased without limit. 

 the expansion of f(.r), found in 19, may Ite, written in the form 



c n , > ln have the meanings assigned in 2:3 find 



f 1 i r 1 i 



- /(*) n (*) 'I* + I - 

 . ) J9 p 



. = 1 1 ' ~ /(') ft (*) *r + f ~ ft W * (*) ' / -'"- 



Jo p Jo p 



